# Who gets credit for the real numbers?

If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more detailed account) routinely get credit for the real numbers?

Stevin did some detailed work (rather than vague general ideas) with unending decimals, including a proof of the intermediate value theorem for polynomials. Newton was in fact inspired by infinite decimals to introduce his general theory of power series.

An interesting point was raised in an answer by Peter Diehr. The so-called Archimedean property (which is one of the defining characteristics of the real number field; though of course it does not suffice to characterize them as the rationals also satisfy it) was considered by authors like Euclid (Elements V.4) considerably earlier. However as far as giving an actual construction (rather than axiomatic definition) Stevin seems to have been the first.

Note 1. To clarify, Stevin developed specific notation for decimals (more complicated than the one we use today) and did actual technical work with them rather than merely envisioning their possibility, unlike some of his predecessors.

Note 2. One useful source for this is Malet, Antoni. Renaissance notions of number and magnitude. Historia Math. 33 (2006), no. 1, 63–81.

Note 3. As Malet notes, "Stevin does not justify his definition" which identifies number and "quantity of anything" because to him the identification is obvious, and the implementation of number is his unending decimals. This was an appropriate move indeed since we know today that the Cantor-Dedekind postulate identifying the number line and the line in physical space is untenable based on what modern physics teaches us; similar remarks apply to magnitude/quantity. Stevin of course was not aware of "transcendental" numbers but no such knowledge is required in order to define the real numbers by means of unending decimals; namely this could have been done even if Liouville did not prove the existence of transcendental numbers.

Note 4. I should clarify that Stevin dealt with unending decimals in his book l"Arithmetique rather than the more practically-oriented De Thiende meant to teach students to work with decimals (of course, finite ones).

Note 5. As far as using the term real to describe the numbers Stevin was concerned with, it should be clarified that the first one to describe the common numbers as real may have been Descartes and at any rate this usage is later than Stevin. On the other hand, if we talk about representing common numbers (including both rational and not so), Stevin not only speculated about the possibility of a representation scheme using decimals, but (unlike some of his predecessors) developed a specific notation (though different from what we use today) and moreover did work with this notation.

Note 6. Cantor thought that Cauchy Completeness (CC) was sufficient to characterize the real numbers axiomatically. Today we know this is not the case, as one also needs the Archimedean property. I found out recently that Dedekind was convinced he had a proof of the existence of an infinite set; see here. Do these misconceptions by Cantor and Dedekind indicate a shortcoming of the constructions of the real numbers they proposed? Hardly so. Stevin's approach to representing all common numbers by unending decimals similarly could not be held at fault because Stevin was not aware of certain future developments.

• Fundamentally, for the first "complete" (i mean: conceptually) theories (axiomatic and/or "constructive") of the real number system, avoiding the geometric intuition, according to which a real number is basically the "numerical representation" of a point on the continuous line. – Mauro ALLEGRANZA May 11 '16 at 9:40
• @Mauro, I am not sure what the noun of the above sentence is. – Mikhail Katz May 11 '16 at 12:45
• @MikhailKatz the noun is implicit: "Fundamentally, [they get credit] for the first..." – KCd May 11 '16 at 13:22
• @Mauro, Stevin didn't need geometic intuitions to work with infinite decimals (though he surely had such intuitions as does every mathematicians). There are some interesting articles in the literature elaborating on the bridge between the discrete and continuous as built by Stevin. – Mikhail Katz May 11 '16 at 16:28
• Well, if we believe Fowler infinite continued fractions were pondered already by Pythagoreans projecteuclid.org/euclid.bams/1183544897 – Conifold May 12 '16 at 6:56

Many people get credit, because this was a long story beginning in the ancient Greece. Euclid has a theory of proportions (based on earlier research) which is equivalent to modern theory of real numbers. Infinite decimal expansions were gradually introduced since 17th century (Napier, Stevin), and the modern theories are due to Cantor and Dedekind. So the development took 2000 years, and it is impossible to credit one person.

• Alexandre, all of mathematics is an organic whole, so all mathematicians should arguably get credit for all of mathematics. Beyond this, it is difficult to attribute unending decimals to Euclid. Stevin did some detailed work with unending decimals, including a proof of the intermediate value theorem for polynomials. Mathematical equivalence is not equivalent to historical equivalence. As far as Napier is concerned, I was not aware of his work on unending decimals. Which Napier are you referring to? As I recall there were two of them. Do you have any details on his work on unending decimal – Mikhail Katz May 11 '16 at 14:03
• "Difficult to attribute infinite decimals to Euclid". The question was not about "infinite decimals" but about real numbers. There are various representations of real numbers, and "infinite decimals" is only one of them. Concerning Napier, I mean John Napier, inventor of the logarithms, and his book where he describes this invention. – Alexandre Eremenko May 11 '16 at 19:38
• As long as we are giving credit I am apalled, apalled I tell you, that Eudoxus isn't mentioned. Euclid just recorded his theory, why does he get all the credit! :) Theory of proportions however could only be applied to already constructed ratios, and geometric constructions came nowhere near modern reals, or even algebraic numbers. – Conifold May 12 '16 at 6:50
• @Conifold: In my answer I was not "giving credit". I only mentioned existing (surviving) sources. And the source for the theory of proportions is Euclid. – Alexandre Eremenko Mar 14 '18 at 17:14
• @Conifold: The source that I mention is Euclid. And I say that is is based on earlier research. I read Euclid, and on my opinion, his theory is equivalent to the modern one. If you have a reference for Eudoxus texts, please give them. And if you disagree with my answer, just write your own. – Alexandre Eremenko Mar 17 '18 at 14:50

The Archimedean property as it is called, was used as an axiom by Archimedes, and he credited Eudoxus of Cnidus, who predates Euclid; also see this.

In fact Stevin does not justify his first definition (“Number is that by which one can tell the quantity of anything”)

So it appears that, like Archimedes, Simon Stevin assumes that every point of a line corresponds to a distance from its origin; that is, magnitudes correspond to points of the line. The nice mathematical distinctions that appear in the 19th century which sort out the details of the Real numbers, are not important to Stevin; what is important is that the decimal notation provides a convenient method for recording these magnitudes.

His work was intended to teach students how to work with decimal numbers. Since even the concept of transcendental numbers does not appear until the 19th century, I don't see how any earlier work could be cited as referring to the Real numbers, except as an axiom.

For reference: Archimedes' Axiom and Archimedean axiom

• Peter, the term "Archimedean property" was introduced by Otto Stolz in 1883. Certainly all these admirable authors predate Stevin but Stevin gave what can be interpreted as an actual construction of the real numbers, namely in terms of unending decimals. If you show me a construction in Archimedes, Eudoxus, or Euclid, I would be very interested, but short of that Stevin seems to be the earliest author to have actually proposed a construction of what have since become known as the oh!-so-real numbers :-) – Mikhail Katz May 11 '16 at 18:16
• The OP didn't mention construction as the goal. Dedekind certainly gave a valid construction, with proofs. The use of decimal numbers, even if unending, is not a constructive proof. And I did not say that Archimedes used that term; I said he used the property as an axiom - a statement which requires no proof. Perhaps you can edit the question to clarify your goal(s). – Peter Diehr May 11 '16 at 22:25
• Meaning of a concept does depend on a available context. Decimals are one of n equivalent ways we define real numbers today, knowing various subkinds of them is part of their conception too. Stevin knew neither the n ways, nor the equivalences, and only little on what kinds of numbers decimals cover. Definition we can prove today is equivalent to a definition of X is not a definition of today's X. Just as Cauchy didn't have thoughts about (modern) continuous functions Stevin didn't have thoughts about (modern) real numbers. Modernizations can be projected back only so far. – Conifold May 15 '16 at 1:39
• @Mikhail I agree that Stevin deserves much more recognition for his influence on the modern conception of number, in particular using infinite decimals to break Euclidean walls between numbers, magnitudes and ratios. My issue is with the phrasing: he did not provide a system for representing real numbers for the same reason that Euclid did not solve quadratic equations, and Eudoxus did not discover a transcendental ratio, those notions weren't available in their time. Cantor and Dedekind can be said, with charity, to have constructed real numbers, but Stevin seems like a bridge too far. – Conifold May 15 '16 at 20:07
• @Mikhail As often, there are no bright lines, nor is there really a need for them to describe how history unfolded, modern labels are tangential to that. l'd look for prototypes of several modern approaches being articulated, and shared sense that they aim at the same notion in different ways. When this happens for real numbers is a blurry judgement call, but the end of 19th is clearly in, and early 17th is clearly out. – Conifold May 16 '16 at 20:02

I've only recently begun reading on the subject of the history of mathematics, and my readings are currently limited to a single text; Boyer's A History of Mathematics. However, according to Boyer, and supported by the wikipedia entry for Simon Stevin, I believe your claim that Stevin dealt with "all" real numbers, "(rational, surd, etc.)", is an overreach.

Quoting Boyer :

Viète, ... , in 1579 had urged the replacement of sexagesimal fractions by decimal fractions. In 1585 an even stronger plea for the use of ten-scale fractions, as well as integers, was made by the leading mathematician in the Low Countries, Simon Stevin of Bruges.

This appears to stop short of claim that Stevin's work was conceptually embracing all real numbers. The linked paper by Malet makes the clear claim that Stevin also considered (some) irrational numbers :

That “any root whatsoever is number” [Stevin, 1585, 8] is also a consequence of identifying numbers and measures

Thus, according to Malet, Stevin does consider algebraic numbers, but again this stops short of claiming that Stevin was in possession of the correct notion of "all real numbers". In other words, although we know now that all real numbers can be represented in this way, it is not clear that Stevin was aware of the true and correct nature of the real numbers and their different types. Perhaps this provides some explanation for why Stevin does not get full credit for the real numbers.

As final point, it may also be worth mentioning that Boyer notes :

It is clear that Stevin was in no sense the inventor of decimal fractions, nor was he the first systematic user of them. More than incidental use of decimal fractions is found in ancient China, in medieval Arabia, and in Renaissance Europe; by the time of Viète's forthright advocacy of decimal fractions in 1579 they were generally accepted by mathematicians on the frontiers of research. Among the common people, however, and even among mathematical practitioners, decimal fractions became widely known only when Stevin undertook to explain the system in full elementary detail.

• Boyer does not distinguish sufficiently between the use of decimal fractions "among common people" as he calls them, on the one hand, and the emphasis on unending decimals in Stevin, on the other. Stevin may not have been the inventor of decimal fractions. Moreover he did not deal with unending decimals in his most famous work, De Thiende. However, in his more specialized work, l'Arithmetique, he does emphasize unending decimals and the idea that all numbers should be representable that way. – Mikhail Katz May 27 '16 at 7:18
• To summarize, nothing either Boyer or Malet say contradictions the contention that Stevin was the first to provide a detailed technical representation of ordinary numbers and unlike his predecessors did work with such representation, which as we know today works for all numbers that we affectionately refer to as being oh!-so-real. – Mikhail Katz May 27 '16 at 7:20
• @MikhailKatz Yes, I take your point regarding "all numbers should be representable that way". But does Stevin have an true understanding of what "all" entails in the case of the reals. It is one possible justification, albeit not very well argued by me. – Nick May 27 '16 at 14:37
• Nick, Cantor had misconceptions about axiomatisations of the real numbers. Thus, he thought that requiring Cauchy completeness was sufficient to characterize the real number system axiomatically. As it turns out this is not the case because one requires also the Archimedean property. Does this imply in any way that his construction of the real numbers was deficient? – Mikhail Katz May 28 '16 at 18:42
• @MikhailKatz That's a very good argument, and I believe I understand the point you are making. Unfortunately my mathematical knowledge lacks the necessary depth to find a meaningful way to counter your argument, assuming such a counter argument exists. In fact, I'm pretty sure no such counter exists in the case of Cantor. – Nick May 28 '16 at 18:53

Man, as a collective, is credited in a famous quote: "God made the integers; all else is the work of man" (or "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"). The quote is often attributed to Leopold Kronecker, see for instance "Philosophies of Mathematics", p. 13, Alexander George, Daniel J. Velleman, 2001. Apparently, the authenticity of the quote is disputed.

This book also kind of credits Dedekind:

Of particular note in this connection is the accomplishment, due primarily to the German mathematician Richard Dedekind (1831-1916), of defining the integers, rationals and reals, taking only the system of natural numbers for granted.

From my education, with a western twist, the Dedekind cut was a construction of real numbers that allowed me to get a instantaneous (perhaps faulty) inner picture of reals, based on (natural) rationals, which I did not grasp before (with unending decimal representations). I learned about it while studying rational (Diophantine) approximations of polynomial roots in the real field and in finite fields (continuing fractions).

In the history of science, the person who gets credits is not always the first. In the West, C. Columbus often gets credits for discovering America, which is probably unfair. Should the first one proving that there were at least one irrational number be credited too?

I begin (after your comments) to think that answers depend of "what kind of real numbers?", in other words, which which structure? As points on a line, as a succession of figures, as a ring or field structure, as a vector space or an an algebra, as a "sense" of continuity?

From my second hand knowledge, some say that arabic/muslim (in a wide sense) mathematicians were the first to treat irrational numbers as algebraic objects (possibly only surds), and indian ones developed trigonometric series (Ideas of Calculus in Islam and India, Katz, 1995). And the first time I heard about (one instance of) the Hamel basis,

a basis for the real numbers \mathbb{R} as a vector space over the field \mathbb{Q} of rational numbers

I understood that my level in mathematics was too narrow to understand what real numbers really were. Since Heron of Alexandria is sometimes credited with the first (western) notion of imaginary numbers, can we expect the real was discovered after the complex?

• Actually this quote is incorrectly attributed to Kronecker directly. A colleague of his named Weber claimed after Kronecker's death that Kronecker said this. I have grave doubts about this because Kronecker would not have used the term "integer". He was almost as suspicious of the negative numbers as he was of transcendental numbers. Furthermore he specifically wrote that the numbers are a creation of the human mind. – Mikhail Katz May 12 '16 at 12:49
• I am not sure what you are trying to say about Dedekind. Everybody knows that he developed a detailed construction of the real numbers. What I am arguing is that there was a fairly adequate construction in the literature considerably before his. Cauchy for one relied on unending decimals without experiencing any need to give an alternative abstract construction thereof. – Mikhail Katz May 12 '16 at 13:05
• Hawking indeed based himself on flawed information about Kronecker when he chose a title for his (Hawking's) book. – Mikhail Katz May 12 '16 at 17:39
• @Mikhail Katz Thank you for the feedback. Being only an amateur in the history of mathematics, I did my best to clarify misty and faulty points – Laurent Duval May 12 '16 at 19:27
• The idea that Kronecker had reservations about negative numbers quickly becomes ridiculous as you start reading his works. – user2255 May 26 '16 at 16:06

The history of decimal point is much older than Simon Stevin.

According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC. But the oldest Chinese book introducing a decimal mark equivalent, is from the thirteenth century.

Around the middle of the tenth century, Al-Uqlidisi wrote "Kitab al-fusul fi al-hisab al-Hindi" (Chapter's book of Indian Arithmetic), which is the earliest surviving book that presents the Indian system.(Survived in a copy of the original which was made in 1157).

In the fourth part of this book Al-Uqlidisi showed how to modify the methods of calculating with Indian symbols, which had required a dust board, to methods which could be carried out with pen and paper. This requirement of a dust board had been an obstacle to the Indian system's acceptance.

Al-Uqlidisi's book is also historically important as it is the earliest known text offering a direct treatment of decimal fractions. Al-Uqlidisi used a decimal dash, above the first digit of the fractional part of the decimal number.( Very simple if compared to Stevin's notation. But similar to the decimal point used by Bartholomaeus Pitiscus).

While the Persian mathematician Jamshīd Al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by Al-Uqlidisi as early as the 10th century.

Rashed puts Al-Kashi's important contribution into perspective. He shows that the main advances brought in by Al-Kashi are:-

(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.

(2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.

(Sorry, I am not permitted to post more than two links.)

http://vedicsciences.net/articles/history-of-numbers-part-2.html

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Uqlidisi.html

• All this is irrelevant because there is no evidence any of these sources intended to use anything like unending decimals. – Mikhail Katz Jun 5 '16 at 6:38
• In his study of hydrostatics De Beghinselen des Waterwichts of 1586, Stevin used what he called “proof by means of numbers,” This approach is similar to a limit, though Stevin did not have the general definition of that concept, and it seems that he did not actually believe in infinite processes. He said that he preferred the ancient Greek approach, and that his method was only an illustration of his results, not a proof. Nonetheless, Stevin’s work helped promote the idea of limits, – AY M Sep 2 '16 at 20:24
• Please see page 173 muhammadalfaridzi.files.wordpress.com/2014/05/… – AY M Sep 2 '16 at 20:33